Question

what is the area of the sector that is not shaded?

\\(12\pi\text{ units}^2\\)
\\(24\pi\text{ units}^2\\)
\\(120\pi\text{ units}^2\\)
\\(144\pi\text{ units}^2\\)

Explanation:

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<reasoning>

Calculate the central angle of the unshaded sector

\[
\theta_{\text{unshaded}} = 360^\circ - 60^\circ = 300^\circ
\]

Calculate the total area of the circle

\[
A_{\text{total}} = \pi \cdot r^2 = \pi \cdot 12^2 = 144\pi
\]

Calculate the area of the unshaded sector

\[
A_{\text{unshaded}} = \frac{300^\circ}{360^\circ} \cdot 144\pi = \frac{5}{6} \cdot 144\pi = 120\pi
\]
</reasoning>

<answer>
<mcq-option>(A) \(12\pi\text{ units}^2\)</mcq-option>
<mcq-option>(B) \(24\pi\text{ units}^2\)</mcq-option>
<mcq-correct>(C) \(120\pi\text{ units}^2\)</mcq-correct>
<mcq-option>(D) \(144\pi\text{ units}^2\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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"Area of a Sector"
]
}
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Answer:

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</pre_analysis>

<reasoning>

Calculate the central angle of the unshaded sector

\[
\theta_{\text{unshaded}} = 360^\circ - 60^\circ = 300^\circ
\]

Calculate the total area of the circle

\[
A_{\text{total}} = \pi \cdot r^2 = \pi \cdot 12^2 = 144\pi
\]

Calculate the area of the unshaded sector

\[
A_{\text{unshaded}} = \frac{300^\circ}{360^\circ} \cdot 144\pi = \frac{5}{6} \cdot 144\pi = 120\pi
\]
</reasoning>

<answer>
<mcq-option>(A) \(12\pi\text{ units}^2\)</mcq-option>
<mcq-option>(B) \(24\pi\text{ units}^2\)</mcq-option>
<mcq-correct>(C) \(120\pi\text{ units}^2\)</mcq-correct>
<mcq-option>(D) \(144\pi\text{ units}^2\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
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